INSTRUMENTAL VARIABLES (Take 1):
CONSTANT EFFECTS
Josh Angrist
MIT 14.387 (Fall 2014)
1
Organizing IV
I tell the IV story in two iterations, first with constant effects,
then in a framework with heterogeneous potential outcomes.
• The constant effects framework focuses attention on the IV solution
for selection bias and on essential IV mechanics
• But first: Why do IV?
• I can’t say "because the regressors are correlated with the errors."
• As we’ve seen, regressors are uncorrelated with errors by definition
• The (short) regression of schooling on wages produces residuals
uncorrelated with schooling (that’s how the good lord made ’em)
• The problem, therefore, must be that the regression you’ve got is not
the regression you want (and that’s your fault!)
2
IV Goes Long
• Suppose the causal link between schooling and wages can be written
fi (s ) = α + ρs + η i
• Imagine a vector of control variables, Ai , called “ability”; write
η i = Ai γ + vi
where γ is a vector of pop. reg. coeffi cients, so vi and Ai are
uncorrelated by construction
• We’d happily include ability in the regression of wages on schooling,
producing this long regression:
Yi = α + ρsi + Ai γ + vi
(1)
The error term here is the random part of potential outcomes, vi , left
over after controlling for Ai
• If E [si vi ] = 0, a version of the CIA, the population regression of Yi
on si and Ai identifies ρ. That’s like saying: "Ai is the only reason
schooling is correlated with potential outcomes."
3
IV and OVB
• IV allows us to estimate the long-regression coeffi cient, ρ, when Ai is
unobserved.
The instrument, Zi , is assumed to be: (1) correlated with the causal
variable of interest, si ; and (2) uncorrelated with potential outcomes
• Here, "uncorrelated with potentials" means Cov (η i ,Zi ) = 0, or,
equivalently, Zi is uncorrelated with both Ai and vi
• This is a version of the exclusion restriction: Zi can be said to be
excluded from the causal model of interest
• Given the exclusion restriction, it follows from equation (1) that
ρ =
=
Cov (Yi , Zi )/V (Zi )
Cov (Yi , Zi )
=
Cov (si , Zi )
Cov (si , Zi )/V (Zi )
”RF ”
”1st”
• The IV estimator is the sample analog of (2)
(2)
4
Two-stage least squares (2SLS)
• In practice, we do IV by doing 2SLS. This allows us to add covariates
(controls) and combine multiple instruments. Returning to the
schooling example, a causal model with covariates is
Yi = α0 Xi + ρsi + η i ,
(3)
where η i is the compound error term, Ai γ + vi . The first stage and
reduced form are
si
Yi
= Xi0 π 10 + π 11 Zi + ξ 1i
= Xi0 π 20 + π 21 Zi + ξ 2i
(4)
(5)
• The reduced form is obtained by substituting (4) into (3):
Yi
= α0 Xi + ρ[Xi0 π 10 + π 11 Zi ] + ρξ 1i + η i
= Xi0 [α + ρπ 10 ] + ρπ 11 Zi + [ρξ 1i + η i ]
= Xi0 π 20 + π 21 Zi + ξ 2i
(6)
5
2SLS Notes
• Again, it’s all about the ratio of RF to 1st:
π 21
=ρ
π 11
In simultaneous squations models, the sample analog of this ratio is
called an Indirect Least Squares (ILS) estimator of ρ
• Where does two-stage least squares come from? Write the first stage
as the sum of fitted values plus first-stage residuals:
si = Xi0 π 10 + π 11 Zi + ξ 1i = ŝi + ξ 1i
2SLS estimates of (3) can be constructed by substituting first-stage
fitted values for si in (3):
Yi =
α 0 Xi + ρŝi + [η i + ρξ 1i ],
(7)
and using OLS to estimate this "second stage" (a version of eq. 6)
• In practice, we let Stata do it: "manual 2SLS" doesn’t get the
standard errors right
6
2SLS example: Angrist and Krueger (1991)
• AK-91 argue that because children born in late-quarters start school
younger, they are kept in school longer by birthday-based compulsory
schooling laws
• There’s a powerful first stage supporting this: Schooling tends to be
higher for late-quarter births; this is driven by high school and not
college, consistent with the CSL story
• The QOB first stage and reduced form are plotted in Figure 4.1.1
• The corresponding 2SLS estimates appear in Table 4.1.1
• 2SLS matches the QOB pattern earnings (the RF) to the QOB pattern
in schooling (the first stage).
• The exogenous covariates include year-of-birth and state-of-birth
dummies, as well as linear and quadratic functions of age in quarters
• QOB Questioned: Bound, Jaeger, and Baker (1995) and Buckles and
Hungerman (2008) argued QOB is correlated with maternal
characteristics. Allowing for this fails to overturn AK conclusions
7
2SLS is a many-splendored thing
• 2SLS is the same as IV where the instrument is ŝi∗ , the residual from
a regression of ŝi on Xi
• One-instrument 2SLS equals IV, where the instrument is z̃i , the
residual from a regression of Zi on the covs, Xi
• One-instrument 2SLS equals indirect least squares (ILS), that is, the
ratio of reduced form to first stage coeffi cients on the instrument. In
other words,
Cov (Yi , ŝi∗ )
V (ŝi∗ )
=
=
Cov (Yi , ŝi∗ )
Cov (si , ŝi∗ )
Cov (Yi , z̃i )
π 21
=
Cov (si , z̃i )
π 11
• With more than one instrument, 2SLS is a weighted average of the
one-at-time (just-identified) estimates (In a linear homoskedastic
constant-effects model, this is effi cient)
8
Multi-Instrument 2SLS (details; mistakes)
• Let
ρj =
Cov (Yi , Zji )
; j = 1, 2
Cov (Di , Zji )
denote two IV estimands using Z1i and Z2i to instrument Di .
• The 2SLS estimand is
ρ2SLS = ψρ1 + (1 − ψ)ρ2 ,
where ψ is a number between zero and one that depends on the
relative strength of the instruments in the first stage.
• Angrist and Evans (1998) use twins and sex-mix instruments
• Using a twins-2 instrument alone, the IV estimate of the effect of a
third child on female labor force participation is -.084 (s.e.=.017).
The corresponding samesex estimate is -.138 (s.e.=.029).
• Using both instruments produces a 2SLS estimate of -.098 (.015).
• The 2SLS weight in this case is .74 for twins, .26 for samesex, due to
the stronger twins first stage.
9
2SLS Mistakes
2SLS . . . so simple a fool can do it . . .
and many do!
What can go wrong?
• As explained in MHE 4.6.1, three mistakes have yet to be relegated to
the dustbin of IV history:
• Manual 2SLS
• Covariate ambivalence
• Forbidden regressions (from the left and the right)
• These can be interpreted as the result of failed attempts to get round
hard-wired 2SLS protocols
• Avoid temptation: let Stata do it!
10
2SLS Lingo
• These terms come to us from simultaneous equations modeling, the
intellectual birthplace of IV:
• Endogenous variables are the dependent variable and the independent
variable(s) to be instrumented; in a simultaneous equations model,
endogenous variables are determined by solving the system
• To treat an independent variable as endogenous is to instrument it, i.e.,
to replace it with fitted values in the 2SLS second stage
• Exogenous variables include covariates (not instrumented) and the
excluded instruments themselves. In a simultaneous equations model,
exogenous variables are determined outside the system
• In any IV study, variables are either: dependent or (other) endogenous
variables, instruments, or covariates
• If you’re unsure what’s what, or find yourself asking variables to play
more than one role . . . seek counseling
11
The Wald estimator
• How were Vietnam-era vets affected by their service?
• Let Di indicate veterans. A causal constant-effects model is:
Yi = α + ρDi + η i ,
(8)
where η i and Di may be correlated. B/c Zi is a dummy,
Cov (Yi , Zi )
= E [Yi |Zi = 1] − E [Yi |Zi = 0],
V (Zi )
with an analogous formula for
ρ=
Cov (Di ,Zi )
.
V ( Zi )
It follows that,
Cov (Yi , Zi )
E [Yi |Zi = 1] − E [Yi |Zi = 0]
=
Cov (Di , Zi )
E [Di |Zi = 1] − E [Di |Zi = 0]
(9)
• A direct route uses (8) and E[ηi |Zi ] = 0:
E[Yi |Zi ] = α + ρE[Di |Zi ]
(10)
Solving this for ρ produces (9)
12
Earnings Consequences of Vietnam-Era Military Service
(Angrist, 1990)
• Key variables
Zi
Di
= randomly assigned draft-eligibility in the 1970-72 draft lotteries
= a dummy indicating Vietnam-era veterans
• The causal effect of Vietnam-era military service is the difference in
average earnings by draft-eligibility status (RF) divided by the
difference in the probability of service (first stage):
Cov (Di , Zi )
V (Zi )
= E [Di |Zi = 1] − E [Di |Zi = 0]
= P [Di = 1|Zi = 1] − P [Di = 1|Zi = 0]
• See RF, first stage, and IV in Angrist (1990), Figures 1-2 and
MHE Table 4.1.3 (based on Angrist 1990,Table 3).
• Draft lottery updates: Angrist, Chen, and Song (2011)
13
Multiple groups and 2SLS
• There’s more to the draft lottery than draft-eligibility: Angrist and
Chen (2008), Figure 1
• Let Ri denote draft lottery numbers. The draft-eligibility Wald
estimator uses 1[Ri < 195] as an instrument in a just-identified setup
• The first stage that uses everything we know can be written:
E [Yi |Ri ] = α + ρP [Di = 1|Ri ],
(11)
since P [Di = 1|Ri ] = E [Di |Ri ]. Suppose Ri ∈ j = 1, ...,J. We can
estimate ρ using J grouped obs by fitting
ȳj = α + ρp̂j + η̄ j
(12)
• Effi cient GLS for grouped data in a constant-effects linear model is
weighted least squares, in this case weighted by the variance of η̄ j
(Prais and Aitchison, 1954). If η i has variance σ2η , the grouped
variance is
σ2η
nj ,
where nj is the group size.
14
Visual IV, Grouping, and GLS
• Equation (12) in action: Angrist (1990), Figure 3. This illustrates
visual instrumental variables (VIV)
• GLS (weighted least squares) applied to equation (12) is 2SLS
• The instruments in this case are dummies for each lottery-number cell.
Define Zi ≡ {rji = 1[Ri = j ]; j = 1, ...J − 1}. The first stage for Di
on Zi plus a constant is saturated, so fitted values are cond. means, p̂j ,
repeated nj times for each j. The second stage slope estimate is
therefore weighted least squares on the grouped equation, (12),
weighted by the cell size, nj
• Because GLS is effi cient, 2SLS is also the effi cient linear combination
of the underlying just-identified IV (Wald) estimates (earlier, we saw
that 2SLS is a weighted average of just-identified estimates in a
two-instrument example)
• That’s why we call Figure 3 "VIV"
15
Specification Testing [TSIV]
Suppose the residuals, ηi , are conditionally homoskedastic. GLS on the
grouped equation, (12), chooses parameter estimates a and b to minimize
ĴN (a, b ) = (1/σ2η ) × ∑ nj (ȳj − a − bp̂j )2
(13)
j
(If the residuals are heteroskedastic, replace constant σ2η with σ2j , the
variance of η i in group j)
• The minimized GLS minimand is the (Sargan) over-id test statistic for
2SLS estimates constructed using group dummies as instruments
(MHE 4.2.2). This test statistic is distributed χ2 (J − 2) if J groups
are used to estimate a slope and intercept
• Over-id for dummy instruments measures the fit of the line
connecting ȳj and p̂j in a VIV plot like Fig. 3 in Angrist (1990)
• The over-identification test statistic is also a (Wald) test statistic for
equality of a full set of linearly independent Wald estimates (implied
by Newey and West (1987); see Angrist (1991))
16
Two-Sample IV
• Let {Yj , Wj , Zj ; j = 1, 2} be data from two samples, where Wj and Zj
Z 0W
include exog covs. Angrist (1990) constructs (first-stage) 2N 2 2 from
military records, while Social Security records were used to construct
Z 0Y
(reduced form) N1 1 1
• AK-95 and Inoue and Solon (2010) simplify: First-stage fits in ds2 are
(Z20 Z2 )−1 Z20 W2 . Carry over to ds1 by constructing the cross-sample
fitted value, Ŵ12 ≡ Z1 (Z20 Z2 )−1 Z20 W2 . The second stage for this
version of TSIV (which AK-95 call SSIV) regresses Y1 on Ŵ12 . The
cross-sample fitted value is
ŵ12,i = Z1i π̂ 2 ,
where π̂ 2 is the first-stage effect estimated using ds2 and the Z1i
(i = 1, ..., N1 ) are the instruments in ds1
• Manual 2SLS, yikes! Inoue and Solon (2010) get the standard errors
right, among other TSIV improvements
17
The Bias of 2SLS
• Cross-section OLS estimates are typically unbiased for the pop BLP,
as well as consistent, but this might not be the regression you want
• 2SLS estimates are consistent for causal effects but biased towards
OLS estimates
• Endogenous var. is vector x; dep. var. is vector y ; no covs:
y = βx + η
(14)
The N ×Q matrix of instruments is Z , with first-stage
x = Z π + ξ
(15)
Outcome error η i is correlated with ξ i . Instruments are uncorrelated
with ξ i by construction and with η i by assumption
• The 2SLS estimator is
�
β2SLS = x 0 PZ x
−1
x 0 PZ y = β + x 0 PZ x
−1
x 0 PZ η
where PZ = Z (Z 0 Z )−1 Z 0 is the projection matrix that produces
fitted values
18
The Bias of 2SLS (cont.)
• Substituting for x in x 0 PZ η, we get
b
�
β2SLS − β =
=
x 0 PZ x
−1
π 0 Z 0 + ξ 0 PZ η
x 0 PZ x
−1
π 0 Z 0 η + x 0 PZ x
(16)
−1
ξ 0 PZ η
(17)
• Expectation of the ratios on the right hand side of (16) are closely
approximated by the ratio of expectations:
�
E [b
β2SLS − β] ≈ E [x 0 PZ x ]
−1
E [ π 0 Z 0 η ] + E [ x 0 PZ x ]
−1
E [ ξ 0 PZ η ] .
This Bekker (1994) approximation ("group asymptotics" in AK-95)
gives a good account of finite-sample behavior
• Using the fact that E [π 0 Z 0 ξ ] = 0 and E [π 0 Z 0 η ] = 0, we have
−1
b
E [�
β2SLS − β] ≈ E π 0 Z 0 Z π + E (ξ 0 PZ ξ )
E ξ 0 PZ η
(18)
• 2SLS is biased b/c E ξ 0 PZ η 6= 0 unless η i and ξ i are uncorrelated
19
The Bias of 2SLS: First-stage F
• Manipulation of (18) generates:
σηξ
E [�
βb2SLS − β] ≈ 2
σξ
E (π 0Z 0Z π ) /Q
+1
σ2ξ
−1
(1/σ2ξ )E (π 0 Z 0 Z π ) /Q is the "population F" for joint significance of
instruments in first-stage, so we can write
σηξ 1
b
E [�
(19)
β
2SLS − β ] ≈
σ2ξ F + 1
• As F gets small, the bias of 2SLS approaches
OLS estimator is
σηξ
,
σx2
which also equals
σηξ
σ2ξ
σηξ
.
σ2ξ
The bias of the
if π = 0. 2SLS estimates
are therefore said to be "biased towards OLS estimates" when there
isn’t much of a first stage. On the other hand, the bias of 2SLS
vanishes when F gets large, as it should happen in large samples
6 0.
when π =
20
The Bias of 2SLS: First-stage F (cont.)
• First-stage F varies inversely with the number of instruments if
they’re weak.
• Adding instruments with no effect on the first-stage R-squared, the
model sum of squares, E (π 0 Z 0 Z π ), and the residual variance, σ2ξ , are
fixed while Q increases
• The F-statistic shrinks as a result. From this we learn that the addition
of weak instruments increases bias
• Holding the first-stage sum of squares fixed, bias is least in the
just-ID case when the number of instruments is as low as it can get
• 2SLS bias is a consequence of first-stage estimation error. We’d like
to use b
xpop = Z π as IVs since these fits are uncorrelated with the
�
second stage error
• In practice, we use b
�
x = PZ x = Z π + PZ ξ, which differs from �
b
xpop by
the term PZ ξ
• 2SLS bias arises from the corr between PZ ξ and η
21
IV without bias or tears
• Just-identified 2SLS (say, the Wald estimator) is approximately
unbiased (this isn’t clear from the Bekker sequence). The just-ID
sampling distribution has no moments, yet it’s approximately centered
where it should be unless the instruments are really weak
• The Reduced Form is unbiased: if you can’t see the relationship
you’re after in the reduced form, it ain’t there! In just-identified
models, the p-value for the reduced-form effect of the instrument is
approximately the p-value from the second stage. (Chernozhukov
and Hansen, 2008, use this to do bias-free inference)
• LIML is approximately median-unbiased for over-identified
constant-effects models, and therefore provides an attractive
alternative to just-identified estimation using one instrument at a
time (see, e.g., Davidson and MacKinnon, 1993, and Mariano, 2001).
(LIML=2SLS in just-identified models)
22
Alternative Estimators
• Riff on the SSIV idea: JIVE (Angrist, Imbens, and Krueger, 1999)
removes bias using leave-out first stage fits for each observation (the
fitted value for observation i is Zi π̂ (i ) where π̂ (i ) is an estimate that
throws out observation i). JIVE sounds appealing, but AIK and
others have found that it rarely beats LIML)
• The right linear combination of OLS and 2SLS should be
approximately unbiased. It turns out that LIML is just such a
"combination estimator" (see the working paper version of AIK-99).
You might also try Fuller’s (1977) modified LIML, discussed by Hahn,
Hausman, and Kuersteiner (2004). Fuller may be more precise than
LIML in finite samples since it has moments
• Hausman, et al. (2007) modify LIML and Fuller to deal with
heteroscedasticity
• Kolesar et al. (2011) modify LIML to allow for random effects
23
Monte Carlo
yi
= βxi + η i
xi
=
Q
∑ πj zij + ξ i
j =1
with β = 1, π 1 = 0.1, π j = 0 ∀j > 1, joint normal errors with
corr (η i , ξ i ) = .8, where the instruments, zij , are independent, standard
normals. The sample size is 1000.
• Figure 4.6.1: OLS, just identified IV (Q=1, labeled IV; F=11.1),
2SLS (Q=2, labeled 2SLS; F=6.0), LIML (Q=2)
• Figure 4.6.2: OLS, 2SLS, and LIML with Q=20 (1 good instrument,
19 worthless; F=1.51)
• Figure 4.6.3: OLS, 2SLS, and LIML with Q=20 but π j = 0;
j = 1, ..., 20 (all 20 worthless; F=1.0)
• Quarter of birth estimates of the returns to schooling (reprise):
Table 4.6.2
24
Tables and Figures
25
A. Average Education by Quarter of Birth (first stage)
13.2
4
13.1
2
3
Years of Education
13
4
4
4
4
3
12.8
4
3
12.7
4
2
4
3
4
4
2
1
3
3
12.9
3
1
2
1
3
2
1
2
1
12.6
1
2
A. Average Education by Quarter of Birth (first stage)
3
12.5
2
1
2
3
1
1
2
12.4
12.3
1
12.2
30
31
32
33
34
35
36
37
38
39
Year of Birth
B. Average Weekly Wage by Quarter of Birth (reduced form)
5.94
5.93
B. Average Weekly Wage by Quarter of Birth (reduced form)
Log Weekly Earnings
5.92
4
3 4
3
1
2
1
2
3
3 4
2
2
5.9
3
3
3
5.91
3
2 3
4
4
2
4
2
4
3
2
2
1
5.89
4
4
4
1
1
1
1
2
1
1
5.88
1
5.87
5.86
30
31
32
33
34
35
36
37
38
39
Year of Birth
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26
Table 4.1.1
2SLS estimates of the economic returns to schooling
OLS
Years of education
Exogenous Covariates
Age (in quarters)
Age (in quarters) squared
9 year-of-birth dummies
50 state-of-birth dummies
Instruments
dummy for QOB = 1
dummy for QOB = 2
dummy for QOB = 3
QOB dummies interacted with
year-of-birth dummies
(30 instruments total)
2SLS
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
.071
(.0004)
.067
(.0004)
.102
(.024)
.13
(.020)
.104
(.026)
.108
(.020)
.087
(.016)
.057
(.029)
Notes: The table reports OLS and 2SLS estimates of the returns to schooling using the Angrist and Krueger (1991)
1980 census sample. This sample includes native-born men, born 1930–39, with positive earnings and nonallocated
values for key variables. The sample size is 329,509. Robust standard errors are reported in parentheses. QOB denotes
quarter of birth.
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27
TABLE 6. IV REGRESSIONS ON RETURNS TO EDUCATION: RESULTS FROM THE CENSUS
Years of Education
Family Controls?
Wages, Logged:
QOB Instruments
0.103
0.147
[0.083]
[0.081]
No
Yes
Wages, Logged:
Year*QOB Instruments
0.075
0.09
[0.040]
[0.040]
No
Yes
Wages, in Levels:
QOB Instruments
33.16
49.16
[24.21]
[23.5]
No
Yes
Wages, in Levels:
Year*QOB Instruments
24.13
30.94
[11.55]
[10.59]
No
Yes
Instruments
QOB
QOB
YOB*QOB
YOB*QOB
QOB
QOB
YOB*QOB
YOB*QOB
Age Controls?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
State Dummies?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Year Dummies?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Weights?
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Notes: Robust standard errors in brackets. Observations are county-of-birth/quarter-of-birth/year-of-birth cells and all regressions weight by total
individuals reporting positive earnings in a cell. The dependent variable in the first two pairs of regressions is the log of average wages in a cell, in the
last two pairs of regressions it is the average of cell wages in levels. Regressions are from cohorts of males born between 1944 and 1960; see Table 5
for a description of family characteristic and wage and age variables.
Courtesy of Kasey Buckles and Daniel M. Hungerman. Used with permission.
28
Courtesy of Joshua Angrist and the American Economic Association. Used with permission.
29
Table 4.1.3
IV Estimates of the Effects of Military Service on the Earnings of White Men born in 1950
Earnings
Mean
Eligibility
Effect
Mean
Eligibility
Effect
Wald
Estimate of
Veteran
Effect
(1)
(2)
(3)
(4)
(5)
1981
16,461
-435.8
(210.5)
1971
3,338
-325.9
(46.6)
1969
2,299
-2.0
(34.5)
Earnings
year
Veteran Status
.267
.159
(.040)
-2,741
(1,324)
-2050
(293)
Note: Adapted from Table 5 in Angrist and Krueger (1999) and author tabulations. Standard errors are shown in
parentheses. Earnings data are from Social Security administrative records. Figures are in nominal dollars. Vete
status data are from the Survey of Program Participation. There are about 13,500 individuals in the sample.
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30
0.15
0.1
0.05
0
‐0.05
‐0.1
‐0.15
‐0.2
‐0.25
estimate
estimate + 1.96*se
‐0.3
estimate ‐ 1.96*se
‐0.35
1970
1973
1976
1979
1982
1985
1988
1991
1994
1997
2000
2003
2006
Figure 1. Draft‐lottery Estimates of Vietnam‐era Service Effects on ln(Earnings) for White Men Born 1950‐52
31
Courtesy of Joshua D. Angrist, Stacey H. Chen, and the American Economic Association. Used with permission.
32
Courtesy of Joshua Angrist and the American Economic Association. Used with permission.
33
0
.25
.5
.75
1
4.7. APPENDIX
0
.5
1
x
OLS
2SLS
1.5
2
2.5
IV
LIML
Figure 4.6.1: Distribution of the OLS, IV, 2SLS, and LIML estimators. IV uses one instrument, while 2S
and LIML use two instruments.
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Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
34
1
.75
.5
.25
0
0
.5
1
x
OLS
LIML
1.5
2
2.5
2SLS
Figure 4.6.2: Distribution of the OLS, 2SLS, and LIML estimators with 20 instruments
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35
1
.75
.5
.25
0
0
.5
1
x
OLS
LIML
1.5
2
2.5
2SLS
Figure 4.6.3: Distribution of the OLS, 2SLS, and LIML estimators with 20 worthless instruments
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36
Table 4.6.2
Alternative IV estimates of the economic returns to schooling
(1)
2SLS
LIML
F-statistic
(excluded instruments)
Controls
Year of birth
State of birth
Age, age squared
Excluded instruments
Quarter-of-birth dummies
Quarter of birth*year of birth
Quarter of birth*state of birth
Number of excluded instruments
(2)
.105
.435
(.020) (.450)
.106
.539
(.020) (.627)
32.27
.42
(3)
(4)
(5)
(6)
.089
(.016)
.093
(.018)
4.91
.076
(.029)
.081
(.041)
1.61
.093
(.009)
.106
(.012)
2.58
.091
(.011)
.110
(.015)
1.97
180
178
3
2
30
28
Notes: The table compares 2SLS and LIML estimates using alternative sets of instruments and controls. The age and age squared variables measure age in quarters. The OLS
estimate corresponding to the models reported in columns 1–4 is .071; the OLS estimate
corresponding to the models reported in columns 5 and 6 is .067. Data are from the Angrist
and Krueger (1991) 1980 census sample. The sample size is 329,509. Standard errors are
reported in parentheses.
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37
MIT OpenCourseWare
http://ocw.mit.edu
14.387 Applied Econometrics: Mostly Harmless Big Data
Fall 2014
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